[Proof of Lemma REF ]
Note that we have the Markov chain \(V\mathrel \multimap \mathrel -{-9mu}\mathrel -X^n\mathrel \multimap \mathrel -{-9mu}\mathrel -Z^n\) . It has been shown in [1]} (see also [2]}, [1]}) that for any channel \(P_{B|A}\)  connecting random variable \(A\)  to \(B\) , we have
\(\eta _{\mathsf {KL}}(P_{B|A}) = \sup _{\begin{array}{c}P_{AU}:\\U\mathrel \multimap \mathrel -{-9mu}\mathrel -A\mathrel \multimap \mathrel -{-9mu}\mathrel -B\end{array}}\frac{I(U; B)}{I(U; A)}.\) 
